nLab framed elliptic curve

Definition

Over the complex numbers, a framed elliptic curve is an elliptic curve equipped with an ordered basis $\{a,b\}$ of the homology of the curve such that the intersection number $a\cdot b$ is 1 (e.g. Hain 08, def. 1.13).

The standard construction of the moduli stack of elliptic curves is in fact via the orbifold quotient of the space of framed elliptic curves acted on by the modular group.

A framing on an elliptic curve is in a way the extreme case of a level structure on an elliptic curve.

A framed elliptic curve is in particular naturally a framed manifold (a framed surface).

Related concepts

• moduli stack of elliptic curves

• moduli space of framed manifolds

References

• Richard Hain, Lectures on Moduli Spaces of Elliptic Curves (arXiv:0812.1803)